The Discrete Ordinates Method for the Neutron Transport Equation in an Infinite Cylindrical Domain

Abstract

We prove a regularity result for a Fredholm integral equation with weakly singular kernel, arising in connection with the neutron transport equation in an infinite cylindrical domain. The theorem states that the solution has almost two derivatives in L1, and is proved using Besov space techniques. This result is applied in the error analysis of the discrete ordinates method for the numerical solution of the neutron transport equation. We derive an error estimate in the L1-norm for the scalar flux, and as a consequence, we obtain an error bound for the critical eigenvalue.

Spatially Varying Discrete Ordinates Methods in Xy-Geometry

“…The problem of optimal regularity for solutions of transport equations is a difficult subject. 6,23 In bounded domains, the solution is not arbitrarily regular, even with smooth coefficients and boundaries. Assuming that φ ∈ H 1 (Ω), the error estimate given here, up to an arbitrarily small factor η, is of the same order as that without coupling.…”

“…The framework and convergence analysis of the sequence of operators T N to T given in their paper will be the starting point of our coupling analysis. Let us also mention that optimal error estimates have been derived by M. Asadzadeh et al 6 in L 1 (Ω).…”

“…More recently the analysis has been extended to the setting of L p spaces, which allow us to obtain more accurate and sometimes optimal error estimates. [4][5][6]18,24 The technical results obtained by Johnson and Pitkäranta 18 will be of important use here. In these works, the number of discrete ordinates was not allowed to depend on the spatial position.…”

Full Error Analysis and Uncertainty Quantification for the Heterogeneous Transport Equation in Slab Geometry

“…Later research e.g. [4], [5], [6] produced analogous results for models of increasing complexity and in higher dimensions, but the proofs were mostly confined to the case of cross-sections that are constant in space. A separate and related sequence of papers (e.g.…”

Introduction: The Radiation Field and its Transfer Equation

“…These schemes are usually referred to as discrete ordinate methods [3]. In particular for large scattering cross sections, these quadrature rules must ensure energy conservation in order to produce even qualitatively correct solutions.…”